Well, not really a newbie, I watched the show back in season 1-3, plus the movies, but totally lost track of it after that.

Well I'm on a nostalgia kick right now. I was wondering about all the later seasons, there are like 15 now, right? How do all the new shows tie in continuity wise, with the original show? Do any of the original characters still appear occassionally?

Continuity continued through season 6. But Characters came and went. There was a big cast change during the middle of Season 5 which replaced the team with a younger group. Which carried into season 6, Power Rangers In Space. Which ended up having a big climax that saw all the evil from the first six seasons get defeated.

Then Season 7, Lost Galaxy started with new characters. It was about a Space Coloney. But there was a big team up with the Space Rangers. And the reformed Main villian returned as Pink Ranger, when the old one "died" ( she wasn't dead by the end).

After that Power Rangers returned to Earth. And every season featured a new cast and storylines. But there was usually a yearly team up. During Wild Force there was a big Team up of all the Red Rangers to celebrate the 10th anniversy. Which featured the return of Tommy Oliver among many.

Production of the show moved to NZ. We haven't gotten a team up every year, since the move. But there have been plenty of nods to the past the last several years. Tommy Oliver returned as a regular for DinoThunder ( only for one season). That season had a clips show about the history of the show, it was the 500 episode. As well as an Episode with Tommy in a coma, where he had to battle himself as Green, White, and Red Zeo Ranger. SPD used some old costumes. Mystic Force featured a cameo by a reformed Rita Repulsa. This season, Operation Overdrive, there was a big team up to celebrate the 15 anniversy. Which featured the return of Adam Park and several Disney era Rangers. Facing off against Zedd and Rita's son.

PR is still in the same continuity , the cast and storylines just change every season. Similar to the Japenese show whre PR gets its action footage from, Super Sentai.

if you havent already, i suggest you watch episode forever red in wildforce, watch the season of dinothunder (which brings back tommy for a full season) and once a ranger which has lord zedd and ritas son in it and 5 rangers from the part team up and take him on, with each season you get a team up from another preveious season which rangerx which would of probably already said but id just thought id say anyway

The series is still in the same universe however once Lightspeed started, they just did their own stuff without any regard to the past. Some seasons do have references to the past so here's a recap of what seasons have references to the past even if it's just teamups.

Ninja Storm = Only 1 I recall is a reference to Triforia but I'm sure there was another 1.

Dino Thunder = This was more reference heavy due to Tommy returning in it so there were some references back to his time.
Also the 6 Ninja Storm along with Sensei, Lothor, Kapri, Marah, Zurgane and Kelzaks.

S.P.D. = Dino Thunder Rangers, 1 ep features just the first 3, 2nd ep features all 5 of them. Also features Zeltrax, T-drones and Triptoids.

Mystic Force = Just Piggy.

Overdrive = Just getting a scale from a dragon in Briarwood which was the city/town that Mystic Force took place in.

Jungle Fury = Nothing.

RPM = Nothing.

Samurai = It only had RPM but it considered RPM to be in another dimension. Only RPM's Ranger Red was in it. Also Bulk being in it and he made a mention of Skull.